Properties

Label 2-684-12.11-c1-0-1
Degree $2$
Conductor $684$
Sign $0.188 - 0.982i$
Analytic cond. $5.46176$
Root an. cond. $2.33704$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.528 − 1.31i)2-s + (−1.44 + 1.38i)4-s − 3.80i·5-s + 3.88i·7-s + (2.57 + 1.16i)8-s + (−4.99 + 2.01i)10-s − 4.83·11-s − 2.15·13-s + (5.09 − 2.05i)14-s + (0.159 − 3.99i)16-s + 5.74i·17-s i·19-s + (5.27 + 5.49i)20-s + (2.55 + 6.34i)22-s − 5.47·23-s + ⋯
L(s)  = 1  + (−0.373 − 0.927i)2-s + (−0.721 + 0.692i)4-s − 1.70i·5-s + 1.46i·7-s + (0.911 + 0.410i)8-s + (−1.58 + 0.636i)10-s − 1.45·11-s − 0.598·13-s + (1.36 − 0.548i)14-s + (0.0399 − 0.999i)16-s + 1.39i·17-s − 0.229i·19-s + (1.18 + 1.22i)20-s + (0.544 + 1.35i)22-s − 1.14·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 684 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.188 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(684\)    =    \(2^{2} \cdot 3^{2} \cdot 19\)
Sign: $0.188 - 0.982i$
Analytic conductor: \(5.46176\)
Root analytic conductor: \(2.33704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{684} (647, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 684,\ (\ :1/2),\ 0.188 - 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.194741 + 0.160871i\)
\(L(\frac12)\) \(\approx\) \(0.194741 + 0.160871i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.528 + 1.31i)T \)
3 \( 1 \)
19 \( 1 + iT \)
good5 \( 1 + 3.80iT - 5T^{2} \)
7 \( 1 - 3.88iT - 7T^{2} \)
11 \( 1 + 4.83T + 11T^{2} \)
13 \( 1 + 2.15T + 13T^{2} \)
17 \( 1 - 5.74iT - 17T^{2} \)
23 \( 1 + 5.47T + 23T^{2} \)
29 \( 1 - 2.51iT - 29T^{2} \)
31 \( 1 - 3.04iT - 31T^{2} \)
37 \( 1 - 0.634T + 37T^{2} \)
41 \( 1 + 7.36iT - 41T^{2} \)
43 \( 1 - 5.91iT - 43T^{2} \)
47 \( 1 + 1.98T + 47T^{2} \)
53 \( 1 - 9.67iT - 53T^{2} \)
59 \( 1 - 10.1T + 59T^{2} \)
61 \( 1 - 5.04T + 61T^{2} \)
67 \( 1 + 0.977iT - 67T^{2} \)
71 \( 1 + 15.0T + 71T^{2} \)
73 \( 1 + 8.86T + 73T^{2} \)
79 \( 1 - 9.39iT - 79T^{2} \)
83 \( 1 - 1.11T + 83T^{2} \)
89 \( 1 + 8.19iT - 89T^{2} \)
97 \( 1 - 2.67T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.54251036145554890404774991567, −9.780779132281479682946510775644, −8.848981130672949686533601293997, −8.457392547420000789184078750239, −7.72979791337535133311696351610, −5.72644365297887175390085636086, −5.16395986107752954445808729528, −4.20973674373003279438765517582, −2.66806034767955993186482608701, −1.69051668111438830847505168672, 0.14502814210252978219296500277, 2.49394364310160631596065182525, 3.79069688601478452703615383555, 4.94261844483501160606024106049, 6.08734394405776813607574323891, 7.07918739432060278731815151527, 7.43109270881511983552885744001, 8.105732176302684283454915889237, 9.863331046848431915322719920481, 10.05640215302836058016616252873

Graph of the $Z$-function along the critical line